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Abstract

Coherent X-ray Diffraction Imaging (CDI) and X-ray ptychography both heavily rely on the high degree of spatial coherence of the X-ray illumination for sufficient experimental data quality for reconstruction convergence. Nevertheless, the majority of the available synchrotron undulator sources have a limited degree of partial coherence, leading to reduced data quality and a lower speckle contrast in the coherent diffraction patterns. It is still an open question whether experimentalists should compromise the coherence properties of an X-ray source in exchange for a higher flux density at a sample, especially when some materials of scientific interest are relatively weak scatterers. A previous study has suggested that in CDI, the best strategy for the study of strong phase objects is to maintain a high degree of coherence of the illuminating X-rays because of the broadening of solution space resulting from the strong phase structures. In this article, we demonstrate the first systematic analysis of the effectiveness of partial coherence correction in ptychography as a function of the coherence properties, degree of complexity of illumination (degree of phase diversity of the probe) and sample phase complexity. We have also performed analysis of how well ptychographic algorithms refine X-ray probe and complex coherence functions when those variables are unknown at the start of reconstructions, for noise-free simulated data, in the case of both real-valued and highly-complex objects.

Fig. 2 Results of reconstructions using both the PC and FC projections for complex valued (max phase= 2π and max phase= 0.5π and) samples at 7 degrees of coherence = σ/L. The reconstruction of the weakly-complex sample has better reconstruction results from the PC projection vis-a-vis the FC projection, but the strongly-complex sample needs higher degree of coherence σ/L for successful convergence. All reconstructions were performed with overlap of 70% in ptychographic simulations. We used a soft-edged square-shape probe for this part of study.

Fig. 3Rreal (FC/PC) Vs. σ/L for 9 σ/L used in simulations, for object maximum phase = 2π and 0.5 π. Two sets of results are done with soft-edged square-shape probe in simulations. This figure is for investigation of cut-off σ/L for maximum phase = 2π and maximum phase = 0.5 π of objects. Our results show that the cut-off σ/L for good reconstructions is higher for max object phase = 2π than for max object phase = 0.5π. In other words, better coherence is needed for highly complex objects. All simulation were performed with 70 % overlap. The horizontal lines drawn here are the cut-off degree of coherence σ/L, above which successful data reconstructions are obtained for both object cases, in the same colour scheme.

Fig. 4 Top: real-space R-factor log-scale FC/PC bonus plots for high phase-diversity (maximum probe phase = 2π) and real-valued probe as a function of degree of coherence σ/L for both object maximum phase = 0 and 2π for 0.3, 0.5 and 0.7 overlap ratios. The horizontal lines in all the graphs in the top panel are cut-off lines for acceptable reconstructions with correctly converged images. Above the cut-off lines the reconstructions are acceptable, while below the lines the reconstructions fail to converge. The performance of a high phase diversity probe is slightly worse than a real probe when overlap ratio is very low (30 % overlap ratio), with 50 % and 70 % overlap ratios the high phase diversity probe always produce better reconstructions results. Bottom: Comparison of real-space R-factors of reconstruction results of high-phase and real-valued probes ratio as a function of degree of coherence σ/L for both object maximum phase = 0 and 2π. Reconstructions are performed with simulated data of 50 % overlap ratio.